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Support varieties for transporter category algebras Fei Xu Department of Mathematics, Shantou University, 243 University Road, Shantou, Guangdong 515063, China

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Article history: Received 12 November 2012 Received in revised form 30 May 2013 Available online 1 August 2013 Communicated by D. Nakano MSC: Primary: 16E40; 16P40; 20C05; 20J06; 55N25 Secondary: 57S17

abstract Let G be a finite group. Over any finite G-poset P we may define a transporter category as the corresponding Grothendieck construction. The classifying space of the transporter category is the Borel construction on the G-space BP , while the k-category algebra of the transporter category is the (Gorenstein) skew group algebra on the G-algebra kP . We introduce a support variety theory for the category algebras of transporter categories. It extends Carlson’s support variety theory on group cohomology rings to equivariant cohomology rings. In the mean time it provides a class of (usually non selfinjective) algebras to which Snashall–Solberg’s (Hochschild) support variety theory applies. Various properties will be developed. Particularly we establish a Quillen stratification for modules. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Let G be a finite group and P a finite G-poset. Throughout this paper, we assume that k is an algebraically closed field of characteristic p, dividing the order of G. We are interested in a finite category G ∝ P , which is the Grothendieck construction on the G-poset P and which we will call a transporter category in this paper. When G = {e} is trivial, {e} ∝ P ∼ = P and when P = • is trivial, G ∝ • ∼ = G. A transporter category G ∝ P is the algebraic or categorical predecessor of the Borel construction EG ×G BP on the finite G-CW-complex BP , in the sense that B (G ∝ P ) ≃ EG ×G BP . Our interests in transporter categories come from the fact that the equivariant cohomology ring H∗G (BP , k) = H∗ (EG ×G BP , k) is Noetherian. Through an algebraic construction of the equivariant cohomology ring, we may introduce in a natural way modules over this ring and hence extend Carlson’s support variety theory for finite group algebras to one for finite transporter category algebras. Let us recall some historical developments in support variety theory. Suppose that X is a compact G-space. Quillen [22,23] proved that H∗G (X ) is Noetherian. Following his notation, we put HG (X ) to be H∗G (X ) if p = 2 or HeGv (X ), the even part of the equivariant cohomology ring, if p ≥ 3. When X = • is just a point fixed by G, the equivariant ring reduces to the group cohomology ring and we shall write H∗G = H∗G (•) and HG = HG (•). Quillen’s work began with the observation that the graded ring HG (X ) is commutative Noetherian. It enabled him to define a homogeneous affine variety VG,X as the maximal ideal spectrum MaxSpec HG (X ), and subsequently described it in terms of VE = VE ,• = MaxSpec HE , where E runs over the set of all elementary abelian p-subgroups of G such that X E ̸= ∅. This is what we nowadays refer to as the Quillen stratification. Restricting to the special case of X = •, based on the fact that Ext∗kG (M , M ) is finitely generated over H∗G ∼ = Ext∗kG (k, k), Carlson [11] extended Quillen’s work by attaching to every finitely generated kG-module M a subvariety of VG = VG,• , denoted by VG (M ) = MaxSpec HG /IG (M ), called the (cohomological) support variety of M, where IG (M ) is the kernel of the following map

φM = − ⊗k M : H∗G ∼ = Ext∗kG (k, k) → Ext∗kG (M , M ). Especially since φk is the identity, VG = VG (k). Following Carlson’s construction, Avrunin and Scott [5] quickly generalized the Quillen stratification from VG to VG (M ). By showing that the support varieties are well-behaved with respect to module

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operations, gradually Benson, Carlson and many others developed a remarkable theory, being a significant progress in group representations and cohomology. Since then, some other analogous support variety theories have been introduced for restricted Lie algebras [16], for finite group schemes [6,17], for complete intersections [4] and for certain finite-dimensional algebras [20,14,24]. Quillen’s work on equivariant cohomology rings has not been fully exploited, partially because there existed no suitable modules which H∗G (X ) (hence HG (X )) acts on or maps to, as in Carlson’s theory. In this article, we attempt to use category algebras to solve the problem: if X = BP comes from a finite G-poset, then we consider the category algebra k(G ∝ P ) of the transporter category G ∝ P , based on which we will generalize Carlson’s theory. In fact, let k be the trivial k (G ∝ P )module (see Section 2.2). Then Ext∗k (G∝P ) (k, k) is a graded commutative ring and there exists a natural ring isomorphism Ext∗k (G∝P ) (k, k) ∼ = H∗ (EG ×G BP , k) = H∗G (BP , k). We shall call the above ring the ordinary cohomology ring of k (G ∝ P ) (instead of the equivariant cohomology ring), as opposed to the Hochschild cohomology ring of k (G ∝ P ). Then we define VG∝P = VG,BP = MaxSpec HG (BP ). The virtue of having an entirely algebraic construction of the equivariant cohomology theory is that it allows us to consider Ext∗k (G∝P ) (M, N) for any finitely generated M, N ∈ k (G ∝ P )-mod, and moreover construct a map

ˆ M : Ext∗k (G∝P ) (k, k) → Ext∗k (G∝P ) (M, M). φM = −⊗ ˆ is the tensor product in the closed symmetric monoidal category (k (G ∝ P )-mod, ⊗, ˆ k). Note that k serves as the Here ⊗ tensor identity. Since we have shown in [29] that Ext∗k (G∝P ) (M, N) is finitely generated over the ordinary cohomology ring, we may define the support variety of M ∈ k (G ∝ P )-mod as VG∝P (M) = MaxSpec HG (BP )/IG∝P (M), where IG∝P (M) is the kernel of φM . Especially VG∝P = VG∝P (k). When P = •, the is exactly Carlson’s construction because k(G ∝ •) ∼ = kG, k ˆ reduces to ⊗k under the circumstance. becomes the trivial kG-module k and ⊗ As a surprising consequence of our investigations of transporter category algebras, we assert that Snashall–Solberg’s (Hochschild) support variety theory (for Gorenstein algebras) applies to every block of a finite transporter category algebra. Furthermore, our support variety theory is closely related with Snashall–Solberg’s as what happens in the case of group algebras and their blocks. A notable point is that the block algebras of a transporter category algebra are usually nonselfinjective and non-commutative, opposing to the cases of (selfinjective) Hopf algebras [11,6,17] and of commutative Gorenstein algebras [4] considered by others. This paper is organized as follows. Section 2 recalls the definitions of the transporter category, the category algebra and the category cohomology. Various necessary constructions are recorded for the convenience of the reader. Here we show a transporter category algebra is Gorenstein and the ordinary cohomology ring of such an algebra is identified with an equivariant cohomology ring. Then in Section 3, we define the support variety for a module over a transporter category algebra. To motivate the reader, we describe the relevant works of Carlson, Linckelmann and Snashall–Solberg, before we develop some standard properties. Sections 4 and 5 contain various properties of support varieties, including the Quillen stratification for modules, as well as results related with sub-transporter categories and tensor products. 2. Preliminaries In this section, we recall the definition of a transporter category and some background in category algebras. Throughout this article we will only consider finite categories, in the sense that they have finitely many morphisms. Thus a group G, or a G-poset P , is always finite. The morphisms in a poset are customarily denoted by ≤. The dimension of a poset P , dim P , is defined to be the maximal integer n such that a chain of non-isomorphisms x0 < x1 < · · · < xn exists in P . Any G-set is usually regarded as a G-poset with trivial relations. One the other hand, since in a G-poset P , both Ob P and Mor P are naturally G-sets, we shall use terminologies for G-sets in our situation without further comments. 2.1. Transporter categories as Grothendieck constructions We deem a group as a category with one object, usually denoted by •. The identity of a group G is always named e. We say a poset P is a G-poset if there exists a functor F from G to sCats, the category of small categories, such that F (•) = P . It is equivalent to saying that we have a group homomorphism G → Aut(P ). The Grothendieck construction on F will be called a transporter category. Definition 2.1.1. Let G be a group and P a G-poset. The transporter category G ∝ P has the same objects as P , that is, Ob(G ∝ P ) = Ob P . For x, y ∈ Ob(G ∝ P ), a morphism from x to y is a pair (g , gx ≤ y) for some g ∈ G. If (g , gx ≤ y) and (h, hy ≤ z ) are two morphisms in G ∝ P , then their composite is easily seen to be (hg , (hg )x ≤ z ) = (h, hy ≤ z ) ◦ (g , gx ≤ y).

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Remark 2.1.2. One can check directly that if HomG∝P (x, y) ̸= ∅ then both AutG∝P (x) and AutG∝P (y) act freely on HomG∝P (x, y). This simple observation is quite useful to us. We note that for each x ∈ Ob (G ∝ P ) = Ob P , AutG∝P (x) is exactly the isotropy group of x. For the sake of simplicity, we will often write Gx = AutG∝P (x), and [x] = Gx, the orbit of x. Note that [x] is a G-set, consisting of exactly the objects in G ∝ P that are isomorphic to x. If P1 is a G1 -poset and P2 is a G2 -poset, then

(G1 × G2 ) ∝ (P1 × P2 ) ∼ = (G1 ∝ P1 ) × (G2 ∝ P2 ). We will utilize this in Section 5.2. From the definition one can easily see that there is a natural embedding ι : P ↩→ G ∝ P via (x ≤ y) → (e, x ≤ y). On the other hand, the transporter category admits a natural functor π : G ∝ P → G, given by x → • and (g , gx ≤ y) → g. Thus we always have a sequence of functors ι

π

P ↩→G ∝ P −→G such that π ◦ι(P ) is the trivial subgroup or subcategory of G. Topologically it is well known that B(G ∝ P ) ≃ hocolimG BP ≃ EG ×G BP . Passing to classifying spaces, we obtain a fibration sequence Bι

Bπ

BP −→EG ×G BP −→BG. Forming the transporter category over a G-poset eliminates the G-action, and thus is the algebraic analogy of introducing the Borel construction over a G-space. This is the first instance explaining why a transporter category has anything to do with the equivariant cohomology theory. Example 2.1.3. If G acts trivially on P , then G ∝ P = G × P . In this case for any x ∈ Ob(G × P ), Gx = G. Example 2.1.4. Let G be a finite group and H a subgroup. We consider the set of left cosets G/H which can be regarded as a G-poset: G acts via left multiplication. The transporter category G ∝ (G/H ) is a connected groupoid whose skeleton is isomorphic to H. In this way one can recover all subgroups of G, up to category equivalences. Making Grothendieck constructions on transitive G-sets reveals the isotropy groups. For an arbitrary G-poset P and x ∈ Ob P , we have a category equivalence G ∝ [x] ≃ Gx , see Remark 2.1.2. For instance, G acts on P = Sp , the poset of non-identity p-subgroup of G, by conjugation. Then Gx = NG (x) for every x ∈ Ob(G ∝ Sp ) = Ob Sp . The isotropy group of x, Gx , is frequently identified with the transporter category Gx × x ∼ = Gx × •. In the upcoming Section 2.2 we will see that a category equivalence D → C induces a Morita equivalence between their category algebras, kD ≃ kC , as well as a homotopy equivalence BD ≃ BC (see [25]). It means there is no essential difference between H and G ∝ (G/H ) as far as we concern. Hence it makes sense if we deem transporter categories as generalized subgroups for a fixed finite group. 2.2. Category algebras and their representations We recall some facts about category algebras. The reader is referred to [25,27,30] for further details. Let C be a finite category and k a field. One can define the category algebra kC , which, as a vector space, has a basis the set of all morphisms in C . We then define a product on the base elements and extend it linearly to kC . The product α ∗β of any two base elements α, β ∈ Mor C is defined to be αβ , if they are composable, or zero otherwise. It is a finite-dimensional associative algebra with identity 1 = x∈Ob C 1x . The category algebra kC possesses a distinguished module k, called the trivial module. It can be defined as k = k Ob C . If α is a base element of kC and x ∈ Ob C , we ask α · x = y if α ∈ HomC (x, y). Otherwise we set α · x = 0. When C is a group, kC is exactly the group algebra and k = k. All modules we consider here are finitely generated left modules, unless otherwise specified. As a convention, throughout this article, the kG-modules are usually written as M , N etc., while the modules of a (non-group) category algebra kC are denoted by M, N etc., except some distinguished modules, namely k and, in the special case of C = G ∝ P , κM which are obtained from kG-modules (to be defined shortly). A k-representation of C is a covariant functor from C to Vectk , the category of finite dimensional k-vector spaces. All representations of C form the functor category VectkC . By a theorem of B. Mitchell (see [25]), the finitely generated left kC -modules are the same as the k-representations of C , in the sense that there exists a natural equivalence VectkC ≃ kC -mod. It is often helpful to utilize the underlying functor structure of a module. For instance, upon the preceding category equivalence we can alternatively define the trivial module k as a constant functor taking k as its value at every object of C . Meanwhile since Vectk is a symmetric monoidal category, VectkC inherits this structure. It means there exists an (internal) tensor ˆ , such that for any two kC -modules M, N, (M⊗ ˆ N)(x) := M(x)⊗k N(x). product, or the pointwise tensor product, written as ⊗ ˆ N via α ⊗ α . Obviously k is the tensor identity with respect to Let α ∈ Mor C be a base element of kC . Then α acts on M⊗

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ˆ and M⊗ ˆN ∼ ˆ M. The category kC -mod has function objects, also called internal hom [27], in the sense that, for ⊗ = N⊗ L, M, N ∈ kC -mod, ˆ M, N ) ∼ HomkC (L⊗ = HomkC (L, H om(M, N)). We record the basic tools for comparing category algebras and their modules. When τ : D → C is a functor between two finite categories, there are adjoint functors for comparing their representations. The functor τ usually does not induce an algebra homomorphism from kD to kC . However it does induce an exact functor, called the restriction along τ , ResC D : kC -mod → kD -mod. If we regard a kC -module as a functor, then its restriction is the precomposition with τ . If we consider the functor π : G ∝ P → G, then any kG-module M restricts to a k(G ∝ P )-module, written as κM = ResGG∝P M, with only one exception k = κk = ResGG∝P k. It is known that H om(κM , κN ) ∼ = κHomk (M ,N ) for any two M , N ∈ kG-mod. In this paper, for the sake of simplicity, if D → C is a functor and M is a kC -module, when it will not cause confusions, we sometimes C neglect ResC D and regard M (instead of ResD M) as a kD -module. The functor ResC is equipped with two adjoints: the left and right Kan extensions along τ D C , RKDC : kD -mod → kC -mod. LKD

The definition of the left and right Kan extensions depend on the so-called over-categories and under-categories, respectively. Despite their seemingly abstract definitions, they are quite computable and thus play an important role in category representations and cohomology, see [26,28,29], as well as Sections 2.5, 3.4 and 4.1. Note that, our notations for the restriction and Kan extensions are slightly different from the earlier articles. The reason is that in this place we feel it is necessary to emphasize the two categories involved in order to make the notations more indicative. 2.3. EI categories When a category C is an EI category, that is, every endomorphism is an isomorphism, there exists a natural partial order on the set of isomorphism classes of objects in C [25]. Indeed the partial order is given by [x] ≤ [y] if and only if HomC (x, y) ̸= ∅. Groups, posets and transporter categories are all EI categories. The upshot is that this partial order allows us to give a filtration of any kC -module M. Let us call x ∈ Ob C an M-object if M(x) ̸= 0. Assume that x is minimal among all M-object, then we can define a submodule Mxˆ such that Mxˆ (y) = M(y) unless y ∼ = x in C , in which case Mxˆ (y) = 0. Then we have a short exact sequence 0 → Mxˆ → M → M/Mxˆ → 0. Denote by Mx = M/Mxˆ . This is an example of the so-called atomic modules of kC . A kC -module M is called atomic if as a functor M takes zero values on all but one isomorphism classes of objects. From the above analysis we see that every M admits a filtration with atomic modules as composition factors. Obviously from any module M and an M-object x we may define an atomic module by brutal truncation (the restriction along the inclusion [x] → C ). Abusing terminology, we always write such modules as Mx . Assume that C is finite EI. Then we can characterize projective and injective kC -modules. Recall from [25] that each indecomposable left (resp. right) projective kC -module, up to isomorphism, is a direct summand of kHomC (x, −) = kC · 1x (resp. kHomC (−, x) = 1x · kC ) for some x ∈ Ob C . The (indecomposable) right (resp. left) injective modules can be obtained as k-duality. Lemma 2.3.1. (1) If P is a projective (or injective) k (G ∝ P )-module, then Px is a projective k(G ∝ [x])-module, for all P-objects x. (2) A k (G ∝ P )-module M is of finite projective (equivalently, finite injective) dimension if and only if Mx is a projective k(G ∝ [x])-module, for all M-objects x. Under the circumstance, both proj.dim M and inj.dim M are bounded by dim P . (3) The transporter category algebra k (G ∝ P ) is Gorenstein, which means, as both the left and the right regular module, it has finite injective dimension. Proof. These are direct consequences of the fact that if HomG∝P (x, y) ̸= ∅, then kHomG∝P (x, y) is both free kGx - and kGy -module, along with the characterizations of projectives and injectives. We only prove (2). Let P∗ → M → 0 be a finite projective resolution. Then its restriction to G ∝ [x] is a finite projective resolution of Mx . Since G ∝ [x] ≃ Gx , their category algebras are Morita equivalent and thus k(G ∝ [x]) is selfinjective. Then the finite projective resolution (Px )∗ → Mx → 0 splits and Mx is a projective (and injective) k(G ∝ [x])-module. On the other hand, assume M satisfies the property that Mx is projective (or zero) for every object x. Take the projective cover P → M. We immediately know that Px → M classes of all minimal x splits as a k(G ∝ [x])-map. Let [y1 ], . . . , [yn ] be the isomorphism n objects among M-objects. Then i=1 nPyi is the projective cover of My which implies i=1 Pyi ∼ = My . Hence if we examine the kernel M′ of P → M, it has the property that M′y is a projective k(G ∝ [y])-module for all objects y such that [y] > [yi ]

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for some 1 ≤ i ≤ n, and zero else. Repeat the same process for M′ , eventually we will obtain the finite projective resolution of M. It implies proj.dim M ≤ dim P . As to the injective dimension, we consider the right k (G ∝ P )-module M∧ (the k-dual of M). It satisfies M∧ (x) = M(x)∧ . Similar to the left module situation, it has a finite projective resolution. When we dualize it, it becomes an injective resolution of M. Since G ∝ [x] ≃ Gx (see Example 2.1.4), in the first two statements, we may replace Px , Mx , k(G ∝ [x]) by P(x), M(x), kGx , respectively. It is helpful to give the following characterization of a transporter category algebra as a skew group algebra. Recall from [3, Chapt. III, Section 4] that if a k-algebra A is a G-algebra, then we may define the skew group algebra A[G] to be the k-vector space A ⊗k kG equipped with a multiplication rule determined by

(a1 ⊗ g1 ) ∗ (a2 ⊗ g2 ) = a1 (g1 · a2 ) ⊗ g1 g2 , where a1 , a2 ∈ A and g1 , g2 ∈ G. Here g1 · a2 means the image of a2 under the action of g1 . For instance we immediately have kG ∼ = (k•)[G]. The reader is referred to [2,3] for further details and some known results about skew group algebras. Proposition 2.2 of [2] says that A[G] is Gorenstein if and only if A is. Since it is easy to verify that kP is Gorenstein, with the following result we have another proof of k (G ∝ P ) being Gorenstein. Lemma 2.3.2. There is an algebra isomorphism k (G ∝ P ) ∼ = kP [G]. Proof. The isomorphism is given by (gx ≤ y, g ) → (gx ≤ y) ⊗ g with inverse (x ≤ y) ⊗ h → (h(h−1 x) ≤ y, h) for x, y ∈ Ob P and g , h ∈ G. The modular representation theory of k (G ∝ P ) will be studied in another place. 2.4. Category cohomology and spectrum For any two kC -modules it makes sense to consider Ext∗kC (M, N) =

i≥0

ExtikC (M, N). Furthermore if M′ and N′ are

ˆ induces a cup product as follows two other modules, the tensor product ⊗ ˆ M′ , N ⊗ ˆ N′ ). ∪ : ExtikC (M, N) ⊗ ExtjkC (M′ , N′ ) → Extik+Cj (M⊗ In particular Ext∗kC (k, k) is a graded commutative ring and we have a natural isomorphism Ext∗kC (k, k) ∼ = H∗ (BC , k) [27]. This ring is called the ordinary cohomology ring of kC and it acts on Ext∗kC (M, N) via the cup product. For any kC -module M, the Yoneda splice provides a ring structure on Ext∗kC (M, M). When M = k, the cup product and Yoneda splice give the same ring structure on Ext∗kC (k, k). There exists a ring homomorphism whose image lies in the center of the graded ring Ext∗kC (M, M)

ˆ M : Ext∗kC (k, k) → Ext∗kC (M, M). −⊗ Moreover given a short exact sequence 0 → M1 → M2 → M3 → 0 the resulting connecting homomorphism is a morphism of Ext∗kC (k, k)-modules. In summary, finite category cohomology behaves very much like the special case of finite group cohomology, except the finite generation property. The ordinary cohomology ring of a category algebra is usually far from finitely generated, but it is so when C = G ∝ P is finite as it is isomorphic to the equivariant cohomology ring H∗G (BP , k), see Section 2.5 and [22,28]. The functor ResC D introduced earlier leads to a restriction on cohomology ∗ ∗ resC D : ExtkC (k, k) → ExtkD (k, k).

It coincides with the restriction H∗ (BC , k) → H∗ (BD , k), induced by the continuous map Bτ : BD → BC between two classifying spaces, see [27]. From now on, we assume that k is algebraically closed. Throughout this paper let us denote by H(C ) = ExtkC (k, k) =

Ext∗kC (k, k), Ext2kC∗ (k, k) ,

if the characteristic of k = 2; if the characteristic of k > 2.

This graded ring is commutative. If Ext∗kC (k, k) is Noetherian, then we can consider the maximal ideal spectrum, a homogeneous affine variety (see [7]), VC = MaxSpec H(C ). Under the circumstance we will call VC the variety of C .

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Assume that both Ext∗kC (k, k) and Ext∗kD (k, k) are Noetherian and there exists a functor τ : D → C . Since the preimage of a maximal ideal is still a maximal ideal, there exists a map between two varieties

ιCD := (resCD )−1 : VD → VC . These varieties and their subvarieties are our main subjects and thus it is helpful if we can handle the restriction. In various interesting cases the map is well understood. As an application of Lemma 2.3.1 we obtain the following result. Lemma 2.4.1. Consider a transporter category G ∝ P and an object x ∈ Ob (G ∝ P ). Then the inclusions Gx × x → G ∝ [x] → G ∝ P induce two restrictions which fit into the following commutative diagram Ext∗k (G∝P ) (M, N)

∝P resG G∝[x]

/ Ext∗kG∝[x] (Mx , Nx ) ∼ =

Ext∗k (G∝P ) (M, N)

∝P resG G ×x

/ Ext∗kG (M(x), N(x)), x

x

for any two modules M, N ∈ k (G ∝ P )-mod. Proof. Let P → M → 0 be a projective resolution. Then Px → Mx → 0 remains a projective resolution of the kG ∝ [x]module Mx , by Lemma 2.3.1(1). Hence the functor P resGG∝ ∝[x] : k (G ∝ P ) -mod → k(G ∝ [x])-mod

induces a restriction P ∗ ∗ resGG∝ ∝[x] : Extk (G∝P ) (M, N) → Extk(G∝[x]) (Mx , Nx ). P Similarly we have a map resGG∝ . The commutative diagram follows directly from the natural isomorphism Ext∗k(G∝[x]) (Mx , x ×x ∗ ∼ Nx ) = ExtkGx (M(x), N(x)).

∗ P P Because the two restrictions resGG∝ : Ext∗k (G∝P ) (M, M) → Ext∗kGx (M(x), M(x)) and resGG∝ ∝[x] : Extk (G∝P ) (M, M) → x ×x Ext∗k(G∝[x]) (Mx , Mx ) are ring homomorphisms with respect to the Yoneda splice, there are two maps P ιGG∝ : VGx ×x → VG∝P , x ×x

induced by resx for M = N = k, and similarly P ιGG∝ ∝[x] : VG∝[x] → VG∝P .

Note that VGx ×x = VG∝[x] . 2.5. Bar resolution and equivariant cohomology One concept that we will refer to is the bar resolution BC ∗ , a combinatorially constructed projective resolution, of k ∈ kC -mod. Given a functor τ : D → C , one can define C∗ (τ /−, k), a complex of projective kC -modules, such that, for each x ∈ Ob C , τ /x is a finite category (the category over x or just an overcategory) and C∗ (τ /x, k) is the (normalized) chain complex resulting from the simplicial k-vector spaces coming from the nerve of τ /x [26]. Since overcategories are used in the proof of a couple of results in the next section, we recall its definition. The objects of τ /x are pairs (d, α) where d ∈ Ob D and α : τ (d) → x is a morphism in C . A morphism from (d, α) to (d′ , α ′ ) is a morphism f : d → d′ in D , satisfying α = τ (f )α ′ . For each finite category D we define BD ∗ = C∗ (IdD /−, k). It is known that C D ∼ LKD B∗ = C∗ (τ /−, k).

As an example, for a group G the only overcategory IdG /• is the Cayley graph and thus BG∗ is the bar resolution of k in group cohomology. For the convenience of the reader we recall from [28] how we obtain equivariant cohomology from category cohomology. Since we can explicitly calculate the unique overcategory as π /• ∼ = IdG / • ×P , it follows that LKGG∝P BG∗ ∝P ∼ = C∗ (π /•, k) ≃ BG∗ ⊗ C∗ (P , k). Thus we have the following chain isomorphisms and homotopy, for any M ∈ kG-mod, Homk (G∝P ) (BG∗ ∝P , κM ) ∼ = HomkG (LKGG∝P BG∗ ∝P , M )

≃ HomkG (BG∗ ⊗k C∗ (P , k), M ) ∼ = HomkG (BG∗ , Homk (C∗ (P , k), M )).

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Note that C∗ (P , k) is a finite complex and consists of each dimension of a permutation kG-module. From [10, VII.7] we see immediately Ext∗k (G∝P ) (k, κM ) ∼ = H∗G (BP , M ) for any M ∈ kG-mod. In [28] we also established the following isomorphism Ext∗k (G∝P ) (κM , N) ∼ = Ext∗k (G∝P ) (k, H om(κM , N)). Since H om(κM , κN ) ∼ = κHomk (M ,N ) , the Ext∗k (G∝P ) (k, k)-action on Ext∗k (G∝P ) (κM , κM ) ∼ = Ext∗k (G∝P ) (k, H om(κM , κM )) can be obtained from the canonical map k → H om(κM , κM ) ∼ = κEndk (M ) , induced by 1k → IdM . This is analogous to the group case. 3. Support varieties for modules Let k be an algebraically closed field. Suppose that C = G ∝ P is a transporter category. We have learned that Ext∗k (G∝P ) (k, k) is Noetherian, over which Ext∗k (G∝P ) (M, N) is a finitely generated module. The action factors through the natural ring homomorphism

ˆ M : Ext∗k (G∝P ) (k, k) → Ext∗k (G∝P ) (M, M), −⊗ and

ˆ N : Ext∗k (G∝P ) (k, k) → Ext∗k (G∝P ) (N, N). −⊗ Based on these, we are about to develop a support variety theory. Since a transporter category G ∝ P is intimately related with G, we will see our theory is a generalization of Carlson’s theory. Standard references for Carlson’s theory include [7, Chapter 5] and [15, Chapters 8–10]. 3.1. Basic definitions For convenience, we shall assume that G ∝ P is connected, unless otherwise specified. It is equivalent to saying that k ∈ k (G ∝ P )-mod is indecomposable or that Ext0k (G∝P ) (k, k) = H0 (B (G ∝ P ), k) = k. It does not mean that P is connected, see Example 2.1.4. Definition 3.1.1. Given a transporter category G ∝ P and modules M, N ∈ k (G ∝ P )-mod, we write IG∝P (M) for the kernel of the map

ˆ M : Extk (G∝P ) (k, k) → Extk (G∝P ) (M, M), −⊗ and VG∝P (M), the support variety of M, for the subvariety MaxSpec(H(G ∝ P )/IG∝P (M)) of VG∝P = VG∝P (k). Since both Ext∗k (G∝P ) (M, M) and Ext∗k (G∝P ) (N, N) act on Ext∗k (G∝P ) (M, N) via Yoneda splice, we further define IG∝P (M, N) as the annihilator of Extk (G∝P ) (k, k) on Ext∗k (G∝P ) (M, N). Then we set VG∝P (M, N) = MaxSpec(H(G ∝ P )/IG∝P (M, N)). We say a subvariety of VG∝P is trivial if it is m = Ext+ k (G∝P ) (k, k), the positive part of Extk (G∝P ) (k, k). Since IG∝P (M, M) = IG∝P (M), we get that VG∝P (M, M) = VG∝P (M), and that IG∝P (M, N) ⊃ IG∝P (M) + IG∝P (N). The latter implies VG∝P (M, N) ⊂ VG∝P (M) ∩ VG∝P (N). Let P be a G-poset and Q an H-poset. Suppose that there exists a group homomorphism φ : H → G as well as a functor θ : Q → P such that φ(h)θ = θ ◦ h for all h ∈ H. For convenience we record such a map as (φ, θ ) : (H , Q) → (G, P ). They induce a functor

Θ : H ∝ Q → G ∝ P, which in turn gives rise to a restriction map P ∗ ∗ resGH∝ ∝Q : Extk (G∝P ) (k, k) → Extk(H ∝Q) (k, k)

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and a map between varieties

ιGH∝∝PQ : VH ∝Q → VG∝P . For instance, Lemma 2.4.1 dealt with (i, i) : (Gx , x) → (G, P ) and (IdG , i) : (G, [x]) → (G, P ), where i stands for the inclusion. Example 3.1.2. Suppose that G acts trivially on P . Then (IdG , pt ) : (G, P ) → (G, •) induce π : G × P → G and resGG×P : Ext∗kG (k, k) → Ext∗k(G×P ) (k, k). Since Ext∗k(G×P ) (k, k) ∼ = Ext∗kG (k, k)⊗Ext∗kP (k, k) by the Künneth formula, and Ext∗kP (k, k) = H∗ (BP , k) is finite-dimensional, VG×P = n VG , where n is the number of connected components of P . The restriction induces a natural map VG×P → VG . With the assumption that G × P is connected, we actually have n = 1 because G × P is connected if and only if P is. At this point, it seems to be a good idea to compare our theory with those of Carlson, Linckelmann and Snashall–Solberg. By putting our approach into the right context, we can better understand the ideas and see what properties we may expect. Afterwards, we will present various results concerning the support varieties. 3.2. Carlson’s theory When P = • our theory is just the theory of Carlson. However, combining recent works in group and category cohomology, Carlson’s theory can be recovered in a more subtle way. To be explicit, if κM ∈ k (G ∝ P )-mod for some M ∈ kG-mod, then we have a commutative diagram [29] Ext∗kG (k, k) resG G ∝P

/ Ext∗kG (M , M )

−⊗M

Ext∗k (G∝P ) (k, k)

ˆ M −⊗κ

resG G ∝P

/ Ext∗k (G∝P ) (κM , κM ).

One can quickly deduce that the restriction map induces Ext∗kG (k, k)/IG (M ) → Ext∗k (G∝P ) (k, k)/IG∝P (κM ), and hence a finite (usually not surjective) map VG∝P (κM ) → VG (M ). If the Euler characteristic χ (P , k) is invertible in k, then by using the Becker–Gottlieb transfer map [29], both vertical maps are injective. Furthermore, if we let P = Sp , the poset of non-identity p-subgroups of G, the left resGG∝Sp becomes an algebra isomorphism (see [10, Chap. X, Section 7] and [13, Part I, Sections 7 and 8] for instance). Hence we get IG (M ) ∼ = IG∝Sp (κM ) and

VG (M ) ∼ = VG∝Sp (κM ).

It means that various properties of VG (M ) can be rewritten for VG∝Sp (κM ). As an example we have a tensor product formula

ˆ N ) = VG∝Sp (κM ⊗N ) = VG (M ⊗ N ) = VG (M ) ∩ VG (N ) = VG∝Sp (κM ) ∩ VG∝P (κN ). VG∝Sp (κM ⊗κ Here the third equality comes from [7, Theorem 5.7.1]. One can similarly deduce other properties for VG∝Sp (κM ) but we shall leave it to the reader as they are just reformulations of known results for VG (M ). Our interests really lie in VG∝Sp (M), or more generally VG∝P (M), for modules M ̸= κM for any M ∈ kG-mod. In the above arguments, Sp may be replaced by various G-subposets which are G-homotopy equivalent to it, see [7, Section 6.6]. A typical example is Ep , the G-poset of all elementary abelian p-subgroups of G. However we want to emphasize that most of our results are established without specifying a poset, see for instance Sections 4 and 5. 3.3. Varieties in blocks

A group algebra can be written as a direct product of (indecomposable) block algebras i bi . (Here for convenience we denote by b, instead of kGb, a block algebra.) Each indecomposable kG-module belongs to exactly one of these blocks in the sense that all but one block act as zero on it. The block that k belongs to is called the principal block, denoted by b0 . In [19], Linckelmann introduced to each block algebra b a Noetherian graded commutative ring H∗ (b), called the block cohomology ring. Then he showed that there exists a natural injective homomorphism ∗ ∗ H (b) → HH (b),

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and thus H∗ (b) acts on Ext∗b (M , M ) via HH∗ (b) = Ext∗be (b, b) (this action is explained in Section 3.4), if M ∈ b-mod. Particularly if b0 is the principal block of a group algebra kG, then H∗ (b0 ) is isomorphic to H ∗ (G, k) and the above injection coincides with the composite of two canonical maps ∗ = H∗ (G, k) → HH∗ (kG) → HH∗ (b0 ). H (b0 ) ∼

Based on these, he was able to define support varieties for modules of a block algebra as above in a natural way [20], being a refinement of Carlson’s theory. Most significantly Linckelmann’s work brought Hochschild cohomology into the theory of support varieties, which was taken up by Snashall and Solberg to develop a new support variety theory using Hochschild cohomology rings, see Section 3.4. Recently [21] Linckelmann has demonstrated that, for a block algebra b, the inclusion H∗ (b) → HH∗ (b) induces an isomorphism upon quotient out nilpotent elements. It implies that the two support variety theories, of Linckelmann based on the block cohomology ring and of Snashall–Solberg defined over the Hochschild cohomology ring of a block of a finite group, are identical. See [8] as well. 3.4. Snashall–Solberg’s theory Snashall and Solberg [24] developed a support variety theory for certain finite-dimensional algebras using Hochschild cohomology rings. Let A be a finite-dimensional algebra, and M , N two finitely generated A-modules. Then there exists a natural action of the Hochschild cohomology ring on Ext groups via the following homomorphism

φM = − ⊗A M : Ext∗Ae (A, A) → Ext∗A (M , M ). Based on Yoneda splice, one can introduce an action on Ext∗A (M , N ) for any two A-modules. For technical reasons, now let A be an indecomposable algebra. Consequently Z (A) becomes a commutative local algebra. Let HH(A) = ExtAe (A, A) be defined in the same fashion as H(C ) in Section 2.4. Assume (Fg.1) there is a graded Noetherian subalgebra H ⊂ ExtAe (A, A) with H0 = Ext0Ae (A, A) = Z (A); and (Fg.2) for any M , N ∈ A-mod, Ext∗A (M , N ) is finitely generated over H. Under the above assumptions, Snashall–Solberg considered the maximal ideal spectrum VH = MaxSpec H. Since H acts on Ext∗A (M , N ) for any two A-modules, assuming IH (M , N ) is the annihilator they defined a subvariety by VH (M , N ) = MaxSpec(H/IH (M , N )). Write IH (M ) = IH (M , M ). Then the (Hochschild) support variety of M ∈ A-mod is given by VH (M ) = MaxSpec(H/IH (M )). They showed that VH (M ) = VH (M , A/ Rad A) = VH (A/ Rad A, M ) and VH (A/ Rad A) = VH . Various satisfactory properties were obtained in [24,14], see Theorem 3.4.4 for a summary. For future reference, we record the following definition. Definition 3.4.1. A subvariety of VH is called trivial if it is ⟨Rad H0 , H+ ⟩, where H+ consists of all the positive degree elements in H. Unfortunately the above conditions (Fg.1) and (Fg.2) are not met by all finite-dimensional algebras, see [26]. Indeed they put restrictions on the algebras that one may consider. For example, two necessary conditions are that the algebra A has to be Gorenstein [14], and that Ext∗Ae (A, A) itself must be Noetherian. Although many algebras do not satisfy (Fg.1) and (Fg.2), we can show that Snashall–Solberg theory works for block algebras of a transporter category algebra k (G ∝ P ). From [28] we know that Ext∗k (G∝P ) (k, k) is a Noetherian graded commutative ring such that Ext∗k (G∝P ) (M, N) is finitely generated over it, for any pair of M, N ∈ k (G ∝ P )-mod. It was also showed there that Ext∗k (G∝P )e (k (G ∝ P ), k (G ∝ P )) is Noetherian. We shall prove that Ext∗k (G∝P ) (k, k)- and Ext∗k (G∝P )e (k (G ∝ P ), k (G ∝ P ))-actions on Ext∗k (G∝P ) (M, N) are compatible and hence it implies that Ext∗k (G∝P ) (M, N) is finitely generated over the Hochschild cohomology ring as well. Theorem 3.4.2. Let M ∈ kC -mod. We have a commutative diagram Ext∗kF (C ) (k, k)

∼ =

Ext∗kC e (kC , kC )

/ Ext∗kC (k, k)

−⊗kC M

ˆM −⊗

/ Ext∗kC (M, M)

with the left vertical map an injective algebra homomorphism.

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This actually generalizes [26, Theorem A]. Let us first recall some other necessary constructions from [26]. For any category C there is a category of factorizations in C , written as F (C ). The objects are the morphisms in C . When a morphism α ∈ Mor C is regarded as an object in F (C ), we will denote it by [α] to distinguish their roles. If [α], [β] are two objects in F (C ), then a morphism [α] → [β] is a pair (µ, γ ), µ, γ ∈ Mor C , such that β = µαγ (that is, α is a factor of β ). When C is a group, F (C ) plays the role of the diagonal subgroup △G ⊂ G × G. Indeed, there is a category equivalence △G ≃ F (G). Given a morphism α in C , we denote by t (α) and s(α) the target and source of α . They induce two functors t : F (C ) → C and ∇ = (t , s) : F (C ) → C e = C × C op , fitting into the following commutative diagram

/ Ce ~ ~ ~~ ~~ p ~ ~ , C

∇

F (C )

CC CC CC CC t !

where p is the projection. By definition t and s send [α] to the target and source of α , respectively. In [26], we investigated e the left Kan extensions LKFC(C ) : kF (C )-mod → kC -mod and LKFC(C ) : kF (C )-mod → kC e -mod, proving LKFC(C ) k ∼ = k and

e ∼ kC . Furthermore LK C induces an isomorphism Ext∗ (k, k) → Ext∗ (k, k), while LK C e induces an injective LKFC(C ) k = kC kF (C ) F (C ) F (C )

algebra homomorphism Ext∗kF (C ) (k, k) → Ext∗kC e (kC , kC ). Especially F (C ) is connected if and only if C is. At last LKCCe : kC e mod → kC -mod is explicitly expressed as LKCCe ∼ = − ⊗kC k, where k is the trivial left kC -module. When M = k, the lower horizontal map becomes the split surjection in [26, Theorem A]. F (C )

F (C )

→ k → 0, and a map f : Bn = Cn (IdF (C ) /−, k) → k representing an element Proof. Consider the bar resolution B∗ e ˆ M ∈ ExtnkC (M, M). We do it by explicit calculations. ξ ∈ ExtnkF (C ) (k, k). We need to prove LKFC(C ) ξ ⊗kC M = LKFC(C ) ξ ⊗ e

e

F (C )

Firstly, LKFC(C ) f : LKFC(C ) Bn

e

= Cn (∇/−, k) → LKFC(C ) k = kC is given on each (x, y) ∈ Ob C e as

e

(LKFC(C ) f )(x,y) : Cn (∇/(x, y), k) → kC (x, y) = kHomC (y, x) by

([α∗ ], (β∗ , γ∗ )) → f[α∗ ] βn αn γn . Here we denote by ([α∗ ], (β∗ , γ∗ )) = ([α0 ], (β0 , γ0 )) → · · · → ([αn ], (βn , γn )) a base element of Cn (∇/(x, y), k), where (βi , γi ) : ∇([αi ]) → (x, y) is a morphism in C e . From ([α∗ ], (β∗ , γ∗ )) we can extract a base element of Cn (IdF (C ) / [βn αn γn ], k), written as [α∗ ] = [α0 ] → · · · → [αn ], so our definition makes sense. Note that β0 α0 γ0 = · · · = βn αn γn . F (C )

Secondly, in a similar fashion, LKFC(C ) f : LKFC(C ) Bn

= Cn (t /−, k) → LKFC(C ) k = k is given on each x ∈ Ob C as

(LKFC(C ) f )x : Cn (t /x, k) → k(x) = k by

([α∗ ], µ∗ ) → f[α∗ ] . Here we denote by ([α∗ ], µ∗ ) = ([α0 ], µ0 ) → · · · → ([αn ], µn ) a base element of Cn (t /x, k) and [α∗ ] the encoded base element of Cn (IdF (C ) /[µn αn ], k). e

e

ˆ M is represented by some (LKFC(C ) f ⊗ ˆ IdM ) ◦ Φn Thirdly, LKFC(C ) ξ ⊗kC M is represented by LKFC(C ) f ⊗kC IdM , while LKFC(C ) ξ ⊗ F (C )

provided that Pn → M → 0 is a projective resolution and Φ∗ : Pn → LKFC(C ) B∗ e F (C ) LKFC(C ) B∗

M. Let us take P∗ = x ∈ Ob C and n ≥ 0 we define

ˆ M is a lifting of the identity map of ⊗

⊗kC M and construct Φ∗ , which has to be unique up to homotopy. To this end, for any

Φnx : 1x · LKFC(C ) BFn(C ) ⊗kC M → 1x · LKFC(C ) BF∗(C ) ⊗ M(x) e

by

([α∗ ], (β∗ , γ∗ )) ⊗ (mw ) → ([α∗ ], β∗ ) ⊗ (βn αn γn ) · my . Here we write each element in M = w∈Ob C M(w) as w mw , in which mw ∈ M(w). Directly from the definition one can e e ˆ IdM ) ◦ Φ . It means LKFC(C ) ξ ⊗kC M = LKFC(C ) ξ ⊗ ˆ M ∈ ExtnkC (M, M). verify LKFC(C ) f ⊗kC IdM = (LKFC(C ) f ⊗ To consider Snashall–Solberg’s theory, we want the algebra in question to be indecomposable. Suppose k (G ∝ P ) = i bi is a decomposition into (indecomposable) block algebras, see for instance [1, Section 13]. Since k (G ∝ P ) is Gorenstein, so are its blocks. Let b be a block of k (G ∝ P ). The above theorem implies that Ext∗b (M, N) is a finitely generated module over the Noetherian ring Ext∗be (b, b), if M, N ∈ b-mod. It means Snashall–Solberg’s theory works perfectly for blocks of finite transporter category algebras.

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Here we record some standard properties from Snashall–Solberg’s theory. For convenience, we write Vb (M, N) = VHH(b) (M, N) if b is a block of k (G ∝ P ) and M, N ∈ b-mod. To be consistent, write Vb (M) = Vb (M, M). Some terminologies are recalled first. Definition 3.4.3. Let A be a finite-dimensional algebra and P∗ → M → 0 the minimal projective resolution of M. Then the complexity of M is cA (M ) = min{s ∈ N r ∈ R

such that dimk Pn ≤ rns−1 , for n ≫ 0}.

Let (−)∧ = Homk (−, k) : kC -mod → kC op -mod be the k-dual functor. Recall that BC ≃ BC op and thus Ext∗kC (k, k) ∼ = Ext∗kC op (k, k). It implies that Ext∗kC (k, k) also acts on Ext∗kC op (N∧ , M∧ ), for any M, N ∈ k (G ∝ P )-mod. Some of these constructions pass to every block algebra of kC . The following statements are taken from Snashall–Solberg [24] and Erdmann–Holloway–Snashall–Solberg–Taillefer [14]. Note that a block of a transporter category algebra is Gorenstein, but in general neither selfinjective nor of finite global dimension. Theorem 3.4.4. Let G be a finite group, P a finite G-poset, and k an algebraically closed field of characteristic p dividing the order of G. Suppose that b is a block of k (G ∝ P ) and M, N are two finitely generated modules of b. Then (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

Vb (M) = S Vb (M, S) = S Vb (S, M), where S runs over the set of all simple b-modules. If 0 → M1 → M2 → M3 → 0 is an exact sequence, then Vb (Mi ) ⊂ Vb (Mj ) ∪ Vb (Ml ) for {i, j, l} = {1, 2, 3}. Vb (M1 ⊕ M2 ) = Vb (M1 ) ∪ Vb (M2 ). Vb (M) = Vb (Ω n (M)) for any integer n such that Ω n (M) ̸= 0. Vb (M, N) = Vbop (N∧ , M∧ ). Particularly Vb (M) = Vbop (M∧ ). dim Vb (M) = cb (M). Vb (M) is trivial if and only if M has finite projective dimension if and only if M has finite injective dimension. Let a be a homogeneous ideal in HH(b) = Extbe (b, b). Then there exists a module Ma ∈ b-mod such that Vb (Ma ) = V (a). If Vb (M) ∩ Vb (N) is trivial, then Extib (M, N) = 0 for i > inj.dim b. If Vb (M) = V1 ∪ V2 for some homogeneous non-trivial varieties V1 and V2 with V1 ∩ V2 trivial, then M = M1 ⊕ M2 with Vb (M1 ) = V1 and Vb (M2 ) = V2 .

Proof. The first five properties follow directly from the definition of support variety for a module of Snashall–Solberg, and can be found in [24]. The rest come from [14, Theorems 2.5, 4.4, Proposition 7.2, Theorem 7.3]. 4. Standard properties of VG ∝P (M) Snashall–Solberg’s theory on a block algebra of a transporter category algebra is quite satisfactory in many ways. However, Hochschild cohomology rings do not behave well comparing with ordinary cohomology rings. For example, since an algebra homomorphism does not necessarily induce a homomorphism between their Hochschild cohomology rings, certain important properties in Carlson’s theory are not expected to exist in Snashall–Solberg’s theory. This is one of the reasons why we believe VG∝P (M) has various advantages over Vb (M) which we try to demonstrate in the rest of this paper. 4.1. Principal block Let k (G ∝ P ) be a transporter category algebra. Remember that we assume G ∝ P is connected, which is equivalent to saying that k is indecomposable. We pay special attention to a special block of the transporter category algebra, closely related to our support variety theory. Definition 4.1.1. Assume that C is a finite connected category. We call a block of kC the principal block if the (indecomposable) trivial module k belongs to it, and consequently name the block b0 . Since one can take a minimal projective resolution of k consisting of projective modules belonging to b0 , Ext∗kC (k, k) = Ext∗b0 (k, k) is an invariant of the principal block, comparable to the group case, see for example [19]. Return to transporter category algebras. We claim (i) Ext∗k (G∝P ) (k, k) is a (usually proper) subring of Ext∗be (b0 , b0 ); 0 (ii) Snashall–Solberg’s theory is valid for the subring

H := ⟨Z (b0 ), Extk (G∝P ) (k, k)⟩ ↩→ Extbe (b0 , b0 ); 0

(iii) Ext∗k (G∝P ) (k, k) ↩→ ⟨Z (b0 ), Ext∗k (G∝P ) (k, k)⟩ induces an isomorphism after quotient out nilpotent elements.

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The ordinary cohomology ring is known to be finitely generated [22,27], and thus so are the rings ⟨Z (b0 ), Ext∗k (G∝P ) (k, k)⟩ and H. Assuming these claims, along with Theorem 3.4.2, it guarantees that Snashall–Solberg’s theory can be implemented to H ⊂ HH(b0 ) for the principal block of k (G ∝ P ). Claim (ii) comes from Theorem 3.4.2. Claim (iii) is easy to verify since the commutative local algebra Z (b0 ) quotients out the unique maximal ideal, that is, its nilradical, is exactly k for k is algebraically closed. To establish Claim (i), we have the following proposition. Proposition 4.1.2. Let C be a finite connected category and b0 the principal block of kC . The injective homomorphism Ext∗kC (k, k) ↩→ Ext∗kC e (kC , kC ) induces an injective homomorphism Ext∗kC (k, k) ↩→ Ext∗be (b0 , b0 ). 0

Proof. As a kC e -module, kC = ⊕i bi . Since LKCCe kC ∼ = k and LKCCe preserves direct sums, we see LKCCe b0 = b0 ⊗kC k = k, and C LKC e bi = 0 if bi is not principal. It implies that the split surjection in [26] Ext∗kC e (kC , kC ) → Ext∗kC (k, k) induced by LKCCe , restricts to a split surjection Ext∗be (b0 , b0 ) → Ext∗kC (k, k). Hence we are done. 0

From the preceding discussions, we get VG∝P (M) = VH (M) for all M ∈ b0 -mod. Note that, for any M ∈ b0 -mod, there exists a finite surjective map Vb0 (M) → VG∝P (M). We do not know yet when it becomes an isomorphism. 4.2. Standard properties A subvariety of VG∝P is trivial if it is the ideal consisting of all positive degree elements of Extk (G∝P ) (k, k). Theorem 4.2.1. Let G be a finite group, P a finite G-poset, and k an algebraically closed field of characteristic p dividing the order of G. Suppose M, N are two finitely generated modules of k (G ∝ P ). Then VG∝P (M) = S VG∝P (M, S) = S VG∝P (S, M), where S runs over the set of all simple k (G ∝ P )-modules. If 0 → M1 → M2 → M3 → 0 is an exact sequence, then VG∝P (Mi ) ⊂ VG∝P (Mj ) ∪ VG∝P (Ml ) for {i, j, l} = {1, 2, 3}. VG∝P (M1 ⊕ M2 ) = VG∝P (M1 ) ∪ VG∝P (M2 ). VG∝P (M) = VG∝P (Ω n (M)) for any integer n such that Ω n (M) ̸= 0. VG∝P (M, N) = V(G∝P )op (N∧ , M∧ ). Particularly VG∝P (M) = V(G∝P )op (M∧ ). dim VG∝P (M) = ck (G∝P ) (M). VG∝P (M) is trivial if and only if M has finite projective dimension if and only if M has finite injective dimension. Let a be a homogeneous ideal in Extk (G∝P ) (k, k). Then there exists a module Ma ∈ k (G ∝ P )-mod such that VG∝P (Ma ) = V (a). (9a) If VG∝P (κHomk (M ,N ) ) is trivial, then Extik (G∝P ) (κM , κN ) = 0 for i > dim P . (1) (2) (3) (4) (5) (6) (7) (8)

(9b) If M, N ∈ b0 -mod and VG∝P (M) ∩ VG∝P (N) is trivial, then Extik (G∝P ) (M, N) = 0 for i > inj.dim b0 . (10a) If χ (P ) ≡ 1 (mod p) and VG∝P (κM ) = V1 ∪ V2 for some homogeneous non-trivial varieties V1 and V2 with V1 ∩ V2 trivial, then κM = κM1 ⊕ κM2 with VG∝P (κM1 ) = V1 and VG∝P (κM2 ) = V2 . (10b) If M ∈ b0 -mod such that VG∝P (M) = V1 ∪ V2 for some homogeneous non-trivial varieties V1 and V2 with V1 ∩ V2 trivial, then M = M1 ⊕ M2 with VG∝P (M1 ) = V1 and VG∝P (M2 ) = V2 . Proof. The first five properties follow directly from the definition of VG∝P (M, N). The proofs are entirely analogous to the group case, see [7,15]. The proof of (6) is exactly the same as that for groups, see for example [7, Proposition 5.7.2]. As for (7), one direction is straightforward. Also by Lemma 2.3.1(2), it is equivalent to saying that M is of finite injective dimension. Now let us assume VG∝P (M) is trivial. Then by (6) ck (G∝P ) (M) = 0. It forces the minimal projective resolution of M to be finite. Part (8) follows from VG∝P (M) = VH (M) if M ∈ b0 -mod. Let a be a homogeneous ideal in Extk (G∝P ) (k, k). Then we can define a homogeneous ideal a′ = ⟨Rad Z (b0 ), a⟩ of H. From Snashall–Solberg’s theory [14, Theorem 4.4], the general form of Theorem 3.4.4(8) for b0 , there exists a b0 -module Ma such that VH (Ma ) = V (a′ ). But V (a′ ) is identified with V (a) under the isomorphism VH → VG∝P , and VG∝P (Ma ) = VH (Ma ). Since (9b) and (10b) are Theorem 3.4.4(9) and (10) specialized to the principal block, we shall prove only (9a) and (10a).

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To prove (9a), we notice that Extik (G∝P ) (κM , κN ) ∼ = Extik (G∝P ) (k, κHomk (M ,N ) ) (see Section 2.5). From (7), the assumption implies that κHomk (M ,N ) has finite injective dimension. Hence the statement follows from Lemma 2.3.1. As to (10a), we recall from Section 3.2 that if χ (P ) ≡ 1 (mod p) then there exists a finite surjective map VG∝P (κM ) → VG (M ) for M ∈ kG-mod. Under the circumstance, let Vi′ be the images of Vi , for i = 1, 2, we see that VG (M ) = V1′ ∪ V2′ with V1′ ∩ V2′ trivial. By [7, Theorem 5.12.1], M = M1 ⊕ M2 satisfying VG (M1 ) = V1′ and VG (M2 ) = V2′ . Then κM = κM1 ⊕ κM2 and hence VG∝P (κM ) = VG∝P (κM1 ) ∪ VG∝P (κM2 ). Moreover since the preceding map between varieties restricts to finite surjective maps VG∝P (κMi ) → VG (Mi ) for i = 1, 2. It implies Vi = VG∝P (κMi ) for i = 1, 2. By Theorem 4.2.1(7) and Lemma 2.3.1, VG∝P (κM ) is trivial if and only if κM has finite projective dimension if and only if M is a projective kGx -module for all x ∈ Ob (G ∝ P ). If P = Ep , the poset of all elementary abelian p-subgroups of G, by Chouinard’s theorem [7, Theorem 5.2.4] it is equivalent to saying that M is a projective kG-module. It is not known yet whether in (9a) and (10a) we may replace the constant modules with arbitrary modules. 4.3. Consequences of module filtrations Recall that since G ∝ P is a finite EI-category, every k (G ∝ P )-module M is constructed from atomic modules Mx , where Mx (y) ∼ = M(x) if y ∼ = x or zero otherwise. The following result says that only M-objects contribute to the variety VG∝P (M). Moreover the non-isomorphisms do not play a big role. P Proposition 4.3.1. We have VG∝P (M) = [x] VG∝P (Mx ) = [x] ιGG∝ ∝[x] VG∝[x] (Mx ), where [x] runs over the set of isomorphism classes of M-objects. P Particularly VG∝P (κM ) = [x] ιGG∝ ∝[x] VG∝[x] (κM ), for any M ∈ kG-mod.

Proof. The containment VG∝P (M) ⊂ [x]∈Is (G∝P ) VG∝P (Mx ) follows from Theorem 4.2.1(2). We establish the equality. Firstly we note that Ext∗k (G∝P ) (Mx , Mx ) ∼ = Ext∗k(G∝[x]) (Mx , Mx ) naturally. Moreover we have a commutative diagram

∝P resG G∝[x]

/ Ext∗k (G∝P ) (M, M)

ˆM −⊗

Ext∗k (G∝P ) (k, k)

Ext∗k(G∝[x]) (k, k)

ˆ Mx −⊗

G∝P resG ∝[x]

/ Ext∗k(G∝[x]) (Mx , Mx )

To establish the above commutative diagram, we can represent cohomology classes by n-fold extensions and notice that ˆ are exact. both restrictions and −⊗− The Ext∗k (G∝P ) (k, k)-action on Ext∗k (G∝P ) (Mx , Mx ) factors through the action by Ext∗G∝[x] (k, k). Hence we have VG∝P (Mx )

∗ ∝P = ιGG∝[ x] VG∝[x] (Mx ). Based on the same diagram we see that IG∝P (M) kills Extk (G∝P ) (Mx , Mx ). It means VG∝P (Mx ) ⊂ VG∝P (M).

We remind the reader that since G ∝ [x] as a category is equivalent to Gx , VG∝[x] (Mx ) can be identified with VGx ×x (M(x)). Corollary 4.3.2. We have VG∝P (M) = and hence VG∝P (κM ) =

G∝P [x] ιGx ×x VGx ×x (M(x)), where [x] runs over the set of isomorphism classes of M-objects,

G∝P [x] ιGx ×x VGx ×x (M )

for any M ∈ kG-mod.

If G acts trivially on a connected poset P , then VG×P (M) = M ∈ kG-mod.

ι

G∝P x G x × x VG

(M(x)). Especially VG×P (κM ) = VG (M ) for any

The reader may go back and have another look at Example 3.1.2. We note that if x ∼ = y in G ∝ P , then there exists an element g ∈ G inducing an isomorphism by conjugation Gx → Gy . It implies that VGx ×x (M(x)) ∼ = VGy ×y (M(y)) ∼ = VG∝[x] (Mx ) and P P P ιGG∝ V (M(x)) = ιGG∝ V (M(y)) = ιGG∝ ∝[x] VG∝[x] (Mx ) = VG∝P (Mx ). x ×x Gx ×x y × y Gy × y P Corollary 4.3.3. If M ∈ k (G ∝ P )-mod and H ⊂ G is a subgroup, then ιGH∝ ∝P VH ∝P (M) ⊂ VG∝P (M). H ∝P P Proof. By the preceding result, we have VH ∝P (M) = x ιH V (M(x)), and VG∝P (M) = x ιGG∝ V (M(x)), where x ×x Hx ×x x ×x Gx ×x x runs over the set of all M-objects. Gx ×x P H ∝P G∝ P Gx × x Since ιGH∝ ∝P ιHx ×x = ιGx ×x ιHx ×x and ιHx ×x VHx ×x (M(x)) ⊂ VGx ×x (M(x)) is known in Carlson’s theory, our claim follows.

In a similar fashion we can show that if Q is a G-subposet of P then P ιGG∝ ∝Q VH ∝Q (M) ⊂ VG∝P (M)

for any M ∈ k (G ∝ P )-mod.

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5. Further properties In this section, we shall deal with comparing varieties of categories. The main results are the generalized Quillen stratification and its consequences. Bear in mind that, for the sake of simplicity, if D → C is a functor and M is a kC -module, when it will not cause confusions, we often regard M (instead of writing ResC D M) as a kD -module, even though its underlying vector space structure usually changes. 5.1. Quillen stratification In [22,23] Quillen worked with G-spaces and equivariant cohomology rings. Here we are interested in G-spaces which are classifying spaces of finite G-posets. In order to make a smooth transition from G-spaces and equivariant cohomology to G-posets and transporter category cohomology, we first recall some of the original constructions and then restrict to our case. Given a G-space X and a H-space Y , a morphism from (H , Y ) → (G, X ) is a pair (φ, F ) such that φ : H → G is a group homomorphism and θ : Y → X is a continuous map satisfying the condition that

φ(h)θ (y) = θ (hy),

∀h ∈ H , y ∈ Y .

Such a morphism induces a continuous map Θ : EH ×H Y → EG ×G X and thus a restriction map between equivariant cohomology G ,X

resH ,Y : H∗G (X ) → H∗H (Y ). One can compare these constructions with those introduced before Example 3.1.2. In Quillen’s papers [22,23], it is proved that if X is a G-space, either compact or paracompact with finite cohomological dimension, then there exists an F -isomorphism

q = qG,X : H∗G (X ) → lim A

← −

p (X )

∗ H (E )

where Ap (X ) is called the Quillen category and its objects will be named Quillen pairs for the G-space X in this article. More explicitly, the objects in Ap (X ) are of the form (E , C ), where E is an elementary abelian p-subgroup of G and C is a (nonempty) connected component of X E . A morphism from (E ′ , C ′ ) → (E , C ) is a pair (φ, i) with φ = cg : E ′ → E for some element g ∈ G and i : gC → C ′ an inclusion. Particularly if g can be chosen to be the identity element of G, then we call (E ′ , C ′ ) a Quillen subpair of (E , C ), and write (E ′ , C ′ ) ≤ (E , C ). In this way, all Quillen pairs form a poset Qp , with obvious G-action. Remark 5.1.1. Another way to construct the Quillen category is to define it as a quotient category of the transporter category G ∝ Qp , where Q is the G-poset of all Quillen pairs for X , through HomAp (X ) ((E ′ , C ′ ), (E , C )) = HomG∝Qp ((E ′ , C ′ ), (E , C ))/CG (E ′ , C ′ ), where CG (E ′ , C ′ ) = {g ∈ G gC ′ = C ′ , ghg −1 = h for all h ∈ E } is called the centralizer of (E ′ , C ′ ). We also denote by NG (E ′ , C ′ ) = G(E ′ ,C ′ ) the normalizer of (E ′ , C ′ ) and WG (E ′ , C ′ ) = NG (E ′ , C ′ )/CG (E ′ , C ′ ) the Weyl group of (E ′ , C ′ ).

Quillen’s map q is induced by (E , •) → (E , C ) → (G, X ), where • is sent into C . Since E acts trivially on C , the choice of an embedding • → C does not matter. In fact in any case this map induces a surjective map H∗E (C ) → H∗E (•) which becomes an isomorphism after quotient out nilpotents in both rings. Consequently MaxSpec HE (C ) = MaxSpec HE (•), that is, VE ,C = VE ,• = VE . Let VG,X = MaxSpec HG (X ). The geometric version of Quillen’s map is V G ,X =

(E ,C )

ιEG,,CX VE ,C =

(E ,C )

ιGE ,,CX ιEE ,,•C VE ,• ,

G ,X E ,C

G ,X

where ι : VE ,C → VG,X is induced by resE ,C . Based on the observation that each morphism (E ′ , C ′ ) → (E , C ) in the category Ap (X ) induces a ring homomorphism H∗E → H∗E ′ , which is compatible with the two maps H∗G (X ) → H∗E ′ and H∗G (X ) → H∗E coming from (E ′ , C ′ ) → (G, X ) and (E , C ) → (G, X ), Quillen then continued to demonstrate that VG,X is a disjoint union of G ,X +

some locally closed subvarieties VE ,C V E ′ = V E ′ ,C ′ J

JJ JJ JJ JJ ιG′,X ′ % E ,C

ιE ′ E

V G ,X

by examining more closely the following diagram

/ VE ,C = VE uu uu u uu G,X zuu ιE ,C .

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E ,C

Here the horizontal map is ιEE ′ (corresponding to H∗E → H∗E ′ ), not the senseless ιE ′ ,C ′ , since a morphism in Ap (X ) is different from the morphisms introduced in the second paragraph of this section. Theorem 5.1.2 (Quillen). The dimension of VG,X equals the maximum p-rank of an elementary abelian p-subgroup from Quillen pairs. We shall come back to Quillen’s results shortly after we sort out all terminologies. Now we turn to the case of X = BP for a finite G-poset P . Let us remind the reader of several relevant results. (a) Suppose a G-space X is either compact or paracompact with finite cohomological dimension. If H∗ (X ) is finitedimensional then so is H∗ (C ) for all possible C from Quillen pairs, [22, Corollary 4.3]. (b) (BP )g = B(P g ) (fixed points) for any g ∈ G, [12, Proposition 66.3]; (c) Let Sp and Ep be the G-posets of non-identity p-subgroups and of elementary abelian p-subgroups of G. Then obviously any elementary abelian p-subgroup E has the property that both (BSp )E and (BEp )E are non-empty. When X = BP for a G-poset P and M ∈ kG-mod, we know (IdG , pt ) : (G, BP ) → (G, •) induces the following restriction map res : Ext∗kG (k, k) = H∗ (G) → H∗G (BP ) = Ext∗k (G∝P ) (k, k), which is injective if the Euler characteristic χ (P ) = χ (BP ) is invertible in the base field, by invoking the Becker– Gottlieb transfer [29]. A typical example of such G-posets is Sp , the poset of non-identity p-subgroups in G, since Brown’s theorem [7, Corollary 6.7.4] says that χ (Sp ) ≡ 1 (mod p). There are several well known subposets that are G-homotopy equivalent to it, and thus possess the same property. Consider the canonical map (IdE , pt ) : (E , X ) → (E , •), where E is an elementary abelian p-group. A more involved result says that the restriction res : H∗E (•) → H∗E (X ) being injective is equivalent to saying that X E is non-empty, [18, IV.1, Corollary 1]. If X = BP such that χ (P ) ≡ 1 (mod p), by [9, Chapter III, Theorem 4.3] we get χ (P E ) ≡ 1 (mod p). This implies that BP E ̸= ∅ and that res is injective, matching the observation we made using the Becker–Gottlieb transfer. From now on we focus on the case X = BP for a finite G-poset P . By doing so, we restrict to a special case of Quillen in order to get rid of topology and unveil the underlying algebraic/categorical constructions. Most importantly we gain the freedom to work with varieties of modules. Let E be an elementary abelian p-subgroup of G. Because BP E = B(P E ), a Quillen pair for the G-space BP is of the form (E , BC ) in which C is a connected component of the poset P E . Thus under the circumstance, we may write each Quillen pair as (E , C ) with C a connected subposet of P E , and call (E , C ) a Quillen pair for the G-poset P . ∝P Quillen stratification in our setting says (to be consistent with terminologies in Section 3.1 we choose to write ιEG× C for

ιGE ,,BBCP and accordingly VG∝P for VG,BP etc.) ∝P ∝ P E ×C VG∝P = ιGE × ιGE × C VE × C = C ιE ×• VE ×• . ( E ,C )

(E ,C )

Note that E ∝ C = E × C is a subcategory of G ∝ P , and VE ×C ∼ = VE ×• . Remark 5.1.3. The theorem of Alperin–Avrunin–Evens [15, Theorem 8.3.1] says that VG ( M ) =

ιGE VE (M ),

E ≤G

where E runs over the set of all elementary abelian subgroups of G and M is a kG-module. Since Gx is the isotropy group of x ∈ Ob P , we know x ∈ Ob P E if and only if E ⊂ Gx . Thus if (E , C ) is a Quillen pair, any object x ∈ Ob C satisfies the condition that E ⊂ Gx . Thus x ×x ιEG× x VE ×x ⊂ VGx ×x = VG∝[x] .

Based on the Alperin–Avrunin–Evens’ theorem, we can rewrite Proposition 4.3.1 and Corollary 4.3.2. VG∝P (M) =

P ιGG∝ ∝[x] VG∝[x] (Mx )

[ x]

=

P ιGG∝ V (M(y)) y × y Gy × y

y

=

∝P ιGE × y VE ×y (M(y))

y,E ⊂Gy

=

(E ,C )

∝P ιGE × C VE ×C (M).

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F. Xu / Journal of Pure and Applied Algebra 218 (2014) 583–601

These shall be useful for us in the sequel. We note that if (E , C ) ∼ = (g E , g C ) for some g ∈ G, that is, two Quillen pairs are G∝P G∝P isomorphic in G ∝ P , then ιE ×C VE ×C (M) = ιg E ×g C Vg E ×g C (M). ′

Let (E , C ) be a Quillen pair and E ′ a subgroup of E. Then we denote by C |E ′ the connected component of P E which contains C , determining a unique Quillen subpair (E ′ , C |E ′ ) ≤ (E , C ). One can easily verify E ′ ×(C |E ′ )

ιEE ′ ιE ′ ×x

C E ×x = ιEE × ×x ιE ′ ×x : VE ′ = VE ′ ×x → VE ×x = VE ,

for every x ∈ Ob C . Lemma 5.1.4. For any Quillen pair (E , C ) and M ∈ k(E × C )-mod

ιEE ′ VE ′ ×(C |E ′ ) (M) = {ιEE ′ VE ′ ×(C |E ′ ) } ∩ VE ×C (M). Proof. By Corollary 4.3.2, VE ×C (M) = Then

ι

E ×C x∈Ob C E ×x VE ×x

(M(x)) because the isotropy subgroup of every x ∈ C is exactly E.

C E ×C E ×x {ιEE ′ VE ′ ×(C |E ′ ) } ∩ ιEE × ×x VE ×x (M(x)) = ιE ×x {[ιE ′ ×x VE ′ ×x ] ∩ VE ×x (M(x))} C E ×x = ιEE × ×x ιE ′ ×x VE ′ ×x (M(x)) E ′ ×(C |E ′ )

= ιEE ′ ιE ′ ×x

VE ′ ×x (M(x))

for every x ∈ Ob C . The second equality uses [7, Proposition 5.7.7]. Consequently we get

{ι

E ′ ′ E ′ VE ×(C |E )

} ∩ VE ×C (M) = {ι

E ′ ′ E ′ VE ×(C |E )

}∩

ι

E ×C E ×x VE ×x

(M(x))

x∈Ob C

=

E ′ ×(C |E ′ )

{ιEE ′ ιE ′ ×x

VE ′ ×x (M(x))}

x∈Ob C

= ιEE ′ VE ′ ×(C |E ′ ) (M). The last equality is true since M(y) = 0 if y ∈ Ob(C |E ′ ) − Ob C , by definition. We set

VE+×C = VE ×C −

ιEE ′ VE ′ ×(C |E ′ )

E ′ $E

VE ×C (M)

+

VEG×∝CP

+

=

VE+×C

∩ VE ×C (M)

∝P = ιGE × C VE × C −

E ′ $E

ιGE ′∝×(PC |E ′ ) VE ′ ×(C |E ′ )

∝P + VEG×∝CP (M)+ = ιGE × C VE ×C (M) .

The following is established by Quillen for arbitrary P and M = k [23] and by Avrunin–Scott for P = • and arbitrary modules [5]. Our generalization is based on both of these special cases. Theorem 5.1.5 (Stratification). Let G ∝ P be a finite transporter category, k an algebraically closed field of characteristic p dividing the order of G and M ∈ k (G ∝ P )-mod. Then VG∝P (M) =

VEG×∝CP (M)+ ,

(E ,C )

where the index runs over the set of isomorphism classes of Quillen pairs. Moreover we have a homeomorphism VE ,C (M)+ /WG (E , C ) ≃ VEG×∝CP (M)+ . Proof. We know for M = k the Quillen stratification VG∝P =

+

VEG×∝CP ,

(E ,C )

with +

VE+×C /WG (E , C ) ≃ VEG×∝CP .

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From Remark 5.1.3 we have VG∝P (M) =

∝P ιGE × C VE ×C (M).

(E ,C )

Since Lemma 5.1.4 implies VE ×C (M)+ = VE ×C (M) −

ιEE ′ VE ′ ×(C |E ′ ) (M),

E ′ $E

we get VG∝P (M) =

∝P + ιGE × C VE × C ( M ) .

(E ,C )

It follows from Quillen’s original result that, if (E , C ) runs over the set of isomorphism classes of Quillen pairs, VG∝P (M) =

VEG×∝CP (M)+ ,

(E ,C )

and that there is a homeomorphism VE ,C (M)+ /WG (E , C ) ≃ VEG×∝CP (M)+ .

5.2. Subcategories and tensor products Here we record a few more expected interesting results as consequences of the Quillen stratification. Theorem 5.2.1. Suppose that P is a G-poset and H is a subgroup of G. Then, for any M ∈ k (G ∝ P )-mod,

ιGH∝∝PP

−1

VG∝P (M) = VH ∝P (M).

P Proof. By Corollary 4.3.3 ιGH∝ ∝P VH ∝P (M) ⊂ VG∝P (M). Given a Quillen pair (E , C ) for H ∝ P , (E , C ) is also a Quillen pair for G ∝ P with WH (E , C ) ⊂ WG (E , C ). By Theorem 5.1.5, there is a commutative diagram

/ / VE ×C (M)+ /WG (E , C )

VE ×C (M)+ /WH (E , C ) ≃

≃

VEH×∝CP (M)+

G∝P ιH ∝P

/ V G∝P (M)+ E ×C

Since the upper horizontal map is surjective and the vertical maps are homeomorphisms, the lower horizontal map is surjective. We are done. We could not find a way to generalize the above statement for an arbitrary sub-transporter category of G ∝ P . However we have the following. Proposition 5.2.2. Suppose that P is a G-poset and Q is a G-subposet. Then, for any M ∈ kG-mod, P ιGG∝ ∝Q

−1

VG∝P (κM ) = VG∝Q (κM ).

Proof. The proof is similar to the previous one. If (E , C ) is a Quillen pair for G ∝ Q, then there exists a unique Quillen pair (E , D ) for G ∝ P . We note that VE ×C (κM )+ = VE ×D (κM )+ . Also by definition if g C = C then g D = D as well, for any g ∈ G. It implies that WG (E , C ) ⊂ WG (E , D ). Recall that for any category D there is a diagonal functor △ : D → D × D . Then image of D is written as △D . If C is a connected poset, then so is C × C . Suppose G ∝ P is a transporter category and (E , C ) is a Quillen pair for the G-poset P . Then (△E , C × C ) is a Quillen pair for the (G × G)-poset P × P . Note that from Remark 2.1.2 (G × G) ∝ (P × P ) ∼ = (G ∝ P ) × (G ∝ P ). Corollary 5.2.3. Suppose that G is a finite group and P is a finite G-poset. If H is a subgroup of G and Q is a H-subposet of P , P then ιGH∝ ∝Q

−1

VG∝P (κM ) = VH ∝Q (κM ). Particularly for any k(G × G)-module N

ιG(G∝×PG)∝(P ×P )

−1

V(G×G)∝(P ×P ) (κN ) = VG∝P (κN ).

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Proof. We apply Theorem 5.2.1 and Proposition 5.2.2 to the following three transporter categories H ∝ Q ⊂ H ∝ P ⊂ G ∝ P. For the special case, we consider G ∝ P ∼ = △G ∝ △P ⊂ △G ∝ (P × P ) ⊂ (G × G) ∝ (P × P ). We need a technical result for the last main result. Proposition 5.2.4. Suppose that D and C are two finite categories with finitely generated ordinary cohomology rings. If M, M′ ∈ kD -mod and N, N′ ∈ kC -mod, then Ext∗k(D ×C ) (M ⊗ N, M′ ⊗k N′ ) ∼ = Ext∗kD (M, M′ ) ⊗k Ext∗kC (N, N′ ). C D C Proof. Let PD ∗ → M → 0 and P∗ → N → 0 be two projective resolutions. Then P∗ ⊗k P∗ → M ⊗k N → 0 is a projective resolution of the k(D × C ) = kD ⊗k kC -module M ⊗k N. The isomorphism follows from C ′ ′ ∼ D ′ C ′ HomkD ⊗kC (PD ∗ ⊗k P∗ , M ⊗k N ) = HomkD (P∗ , M ) ⊗k HomkC (P∗ , N ),

and the Künneth formula. When we examine transporter categories, the above result has the following consequences. Corollary 5.2.5. Let G1 ∝ P1 and G2 ∝ P2 be two transporter categories. Then there is a natural isomorphism V(G1 ×G2 )∝(P1 ×P2 ) = V(G1 ∝P1 )×(G2 ∝P2 ) = VG1 ∝P1 × VG2 ∝P2 . If M ∈ kG1 -mod and N ∈ kG2 -mod, under the above isomorphism we have furthermore V(G1 ×G2 )∝(P1 ×P2 ) (κM ⊗N ) = V(G1 ∝P1 )×(G2 ∝P2 ) (κM ⊗k κN ) = VG1 ∝P1 (κM ) × VG2 ∝P2 (κN ). Proof. From Remark 2.1.2 we have (G1 × G2 ) ∝ (P1 × P2 ) ∼ = (G1 ∝ P1 ) × (G2 ∝ P2 ). The statements on varieties are true, because Ext∗k[(G1 ×G2 )∝(P1 ×P2 )] (κM ⊗N , κM ⊗N ) ∼ = Ext∗k[(G1 ∝P1 )×(G2 ∝P2 )] (κM ⊗ κN , κM ⊗ κN )

∼ = Ext∗k(G1 ∝P1 ) (κM , κM ) ⊗ Ext∗k(G2 ∝P2 ) (κN , κN ). Based on the preceding results, we can extend the tensor product formula VG (M ⊗k N ) = VG (M )∩ VG (N ) of Avrunin–Scott [5].

ˆ N) = Theorem 5.2.6. Let G be a finite group and P a finite G-poset. Suppose that M , N are two kG-modules. Then VG∝P (κM ⊗κ VG∝P (κM ) ∩ VG∝P (κN ). Proof. Let us consider the functor △ : G ∝ P → (G ∝ P ) × (G ∝ P ) and the restriction along it. Because

ˆ N = Res(GG∝∝PP ) × (G∝P ) (κM ⊗ κN ), κM ⊗κ by Corollary 5.2.3 we have (G∝P ) × (G∝P ) −1

ˆ k κN ) = ιG∝P VG∝P (κM ⊗

V(G∝P ) × (G∝P ) (κM ⊗k κN ).

From (G × G) ∝ (P × P ) ∼ = (G ∝ P ) × (G ∝ P ) and Corollary 5.2.5 the right-hand-side is

ι(GG∝∝PP ) × (G∝P )

−1

{VG∝P (κM ) × VG∝P (κN )},

which exactly is VG∝P (κM ) ∩ VG∝P (κN ). With Theorem 5.2.6, taking Theorem 4.2.1(5) in account one can see there is some relation between the conditions in (9a) and (9b) in Theorem 4.2.1. Acknowledgments The author was partially supported by a Beatriu de Pinós research fellowship from the government of Catalonia of Spain, as well as a Grant MTM2010-20692 ‘‘Analisis local en grupos y espacios topologicos’’ from the Ministry of Science and Innovation of Spain. References [1] J. Alperin, Local Representation Theory, in: Cambridge Studies in Adv. Math., vol. 11, Cambridge University Press, 1986. [2] M. Auslander, I. Reiten, Cohen–Macaulay and Gorenstein Artin algebras, in: Representation Theory of Finite Groups and Finite-Dimensional Algebras, in: Progress in Math, vol. 95, Birkhäuser, 1991, pp. 221–245.

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